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Professor Irving Michelson

Scientifically speaking. . .

Dr.  Michelson is professor of mechanics, Illinois Institute of Technology, Chicago.

19-Year Lunar Calendar Cycle: Accurate Adjustment to 365 1/4-Day Civil Calendar

Hard Science: When you can state that the lunar month is precisely 29.530589 days long, with eight-digit precision--equivalent to certainty to within one part in one hundred million--that's hard science!

Astronomical Stability: When the Earth's rotation around its polar axis and orbital motion around the Sun, and the Moon's orbital motion around the Earth, are so constant over such a long period as to permit eight-digit precision to be determined by observation--and eclipse cycle periodicities to be known "from antiquity"--can you blame the astronomers for believing in the order and in the stability of our solar system?

Pensée readers are accustomed to situating events in time according to New Moon or other lunar phases.  They have also seen suggestions that Earth movements--rotational or orbital, or both--relative to the Sun and the heavens might have been quite different in historical times than they are now, by reason of different year lengths of 354, or 360, or other numbers of days.  It is interesting to see how these questions are pulled together with special overtones relative to catastrophism--by drawing on only one datum of observational astronomy and by invoking mathematics at the level of elementary arithmetic only.  The accurate adjustment of the 19-year lunar calendar cycle to the 365 1/4-day civil calendar is thereby fully explained.

Synodic Month and Civil Year

The mean synodic month (syzygy period separating one New Moon occurrence from the next) is accurately known to be given by

M = 29.530589 days

with eight-significant-figure (i.e., eight digit) accuracy, implying certainty to within one part in a few hundred millions.  Whether such precision may any longer be regarded as mind-boggling or not, long-term observational determination that permits the quantity to be so fully specified as the number 29.530589 gives it, stands as an eloquent tribute to the "hard sciences" at their best.  Long before the exact length of the year was determined, it was known that one synodic revolution of the Moon is accomplished in about 291/2 days.  The lunar month everywhere served as the basic time element for calendar purposes.

Julius Caesar abolished the use of the lunar calendar and regulated the civil year entirely by the Sun.  The civil year Y, which he took as by intercalating one leap year (of 366 days) each fourth year (the other years having 365 days), is also our common civil calendar year.  Differing slightly from the true solar astronomical ("tropical") year taken as the interval between successive vernal equinoxes, it generates a cumulative error that amounted to roughly 10 days by the middle of the sixteenth century.  The adjustment leading to our present civil calendar ("Gregorian") retains the 365.25 day mean value throughout each century, but eliminates the turn-of-the-century leap year for three out of every four centuries.  The year 2000 will thus be the first turn-of-the-century leap year of the Gregorian calendar, as 1900, 1800 and 1700 were all ordinary years.

Strictly speaking, the number 365.25 has absolutely no astronomical significance apart from its approximate equivalence to the solar (or "tropical") year now known accurately as 365.2421988 days.  Suppressing three leap years in every period of 400 years, according to the Gregorian rule, compensates largely for the error introduced by taking each year as either precisely 365 days or as 366 days in the manner indicated.  The astronomically-based lunar calendar universally employed before Roman times, and widely followed by large segments of the non-Western world since then, is also capable of counting the years perfectly well when suitably adjusted.  Even though the numerical values of M and Y given above do not stand in any exact rational fraction ratio to each other using small enough integers to be directly useful, rules can be found that serve to adjust lunar and solar calendars to each other in one fashion or another.  Simpler rules lead to greater errors, more elaborate ones give smaller errors in return for the inconvenience the greater complexity introduces.  Practical purposes are served best by establishing a periodic cycle of the smallest number of years that agrees accurately with a whole number of lunar months.  Arbitrary judgment decides how to strike the compromise between accuracy and convenience of simplicity associated with shortness of period.

Calendar Cycles

To introduce the meaning of calendar cycles and demonstrate why the smallest number of years is preferred, it is useful to consider a simpler calendar question.  Start by noting, for example, that October 24, 1974, fell on a Thursday--and ask when the same date will again fall on the same weekday, Thursday, in future.  The next time will be in 1985, 11 years later; but it is not correct to conclude that the interval is always 11 years.  For different initial dates it may be five years or six years.  What is certain, however, is that the particular interval specified by the long solar cycle, containing just 28 years, will assure the desired weekday recurrence.  The usefulness of this cycle lies in its correctness for any starting date, provided only that the exceptional end-of-century non-leap-years are avoided.  The rule applies to any dates included in the time span from 1901 to 2099 and separated by 28 years from each other.  Longer periods of 56 (= 2 x 28) years, 84 (= 3 x 28) years, etc., also share the same property since these are simple multiples of the fundamental period of 28 years.  That period is thus most basic and most useful--a 1947 pretty-girl calendar can be dusted off and used again in 1975, for example.

Months, Years, Set Feasts--and Calendar Adjustment

Returning now to the lunar calendar, even the most primitive minds must have early recognized that the four seasons complete their cyclic march of progression in an interval of roughly 12 New Moons, i.e., in 12 synodic months of about 29 ½ days each.  They might reasonably take the season cycle (year) as 12 lunar months-giving a year of 354 days.  One year counted from a particular date would be succeeded by another 354 days later, and the next year would then start at an earlier part of the season than in the previous year.  New Year day would slide around the entire cycle of seasons in roughly 32 years (»365/11 1/4).

Even if people might find some enjoyment in the novelty of New Year's celebrations progressing through all seasons, religious constraints reinforced other preferences for seasonal regularity--and forced adjustments of the lunar calendar.  Add a month now and then, for instance.

In the Hebrew Bible, specifically, the feast of the harvest of "the first-fruits of thy labours, which thou sowest in the field" is prescribed to fall "On the fifteenth day of this seventh month. . ." (Exodus 23:16; Leviticus 23:34).  Harvest of first fruits locks in to the seasons; fixing the month of its occurrence forces an adjustment of a strictly lunar 12-month year.  Occasional 13-month years do it.

Partial Fractions--Origin of Nineteen-Year Cycle

When the 12-month year is recognized to be too crude because of the error of 11 days accumulating from one year to the next, the question presents itself whether a somewhat longer (but not too long) period of a whole number of years would be better.  Specifically, with a civil year of 365 1/4 days, the question is to discover what whole number of these years most accurately corresponds to a different whole number of lunar months.  Then New Moon occurring once on a given civil date will assuredly occur on the same civil date in the earlier and later years specified by the cycle so obtained.

The required calculation depends on just two quantities, the year Y and the month.  M already quoted as

Y = 365.25 days

M = 29.530589 days

It is because the ratio of these two numbers is just more than 12 that there are 12 full lunar months in each year.  There also remain about 11 1/4 days, as earlier noted.

Closer and more useful matching than 12 lunar months per one year requires more precise evaluation of the ratio of periods Y and M.  On the other hand, the refinement sought is much less than the eight-digit precision of M implies.  With no more than two-decimal accuracy for M, in fact, an adequately improved evaluation proceeds by examining the ratio in proper fraction form as

Y       36525              1089
--  =  -------  =  12 + ------
M      2953                2953

and carried out to show 12 whole months and the fractional remainder as indicated.  A first step improvement of the calendar analyzed most crudely as a 12-month year proceeds by noting that the remainder is roughly 1/3 in value.  Since 12 plus 1/3 is equivalent to 37/3, we see that 3 years of 365 1/4 days are now approximated by the whole number 37 lunar months.  The discrepancy is found by comparing the number of days in each.

3 x 365.25 = 1,095.75 days

37 x 29.53 = 1,091.61 days.

There is now an error of only 4.14 days in three years.  That is still more than one day every year, and although it is a great improvement compared with the year consisting of 12 lunar months, each new year would regress in season and travel around the whole four seasons in about 266 years.  It is easy to do much better by carrying out the calculation just one more step in the same manner that has led to the 37-month cycle.

For this purpose it is only necessary to evaluate the ratio 1089/2953 more precisely than as 1/3.  Hence note that its reciprocal is not 3 but

2953 =         775
-----  = 2 + -------
1089            1089

Now the remainder fraction is seen to be nearly 5/7 in value (error of less than one half of one percent of this value), and 2 plus 5/7 is 19/7--more accurately than the value 3 adopted earlier.  Finally, then,

Y                  1                     7
--   = 12 +  ------   = 12 +  -- 
M                19/7                19

      =  -----

In this approximation, we have 19 years of 365 1/4 days in a period of 235 lunar months M days in length each month.  Using the accurate value of M given earlier, we evaluate the present approximation by comparing as before and find

19 x 365.25 = 6,939.75 days

235 x 29.530589 = 6,939.6884 days.

The discrepancy is now only 0.0616 day, or less than one hour and a half, in 19 years.  It accumulates to one day only in 303 years.

The 19-year cycle just demonstrated is nothing new.  Athenian astronomer Meton announced its discovery in year -432.  His discovery of the 19-year cycle presupposes precise knowledge of the length of the lunar month as well as of the solar (tropical) year of 365.2421988 days, to second--decimal accuracy at least.  This is seen by taking the lunar month as 29 ½ days rather than 29.53 and observing that it leads not to a 19-year cycle, but to one of 21 years.  Implications for planetary catastrophism in historical times are then rather direct.

Can it be imagined that the Earth's orbital and/or rotational motion might have been severely disrupted a thousand years before Meton?  One or the other would entail a change in year length that would certainly affect the value of the second decimal.  Could Earth's motion, and that of Moon also, so quickly stabilize to present values following catastrophic encounters with bodies like proto-Venus or Mars, as to enable Meton 2500 years ago to have the precise values required to determine the 19-year cycle?  The implications are not positive: It is easier to imagine that years of length 354 days, or 360, might merely have been adopted for convenience of calendar determination either with New Year shifting through seasons or with intercalated months as needed to provide adjustment assuring no such shifts.  Why not think of near-encounters that did not make the Earth "stand still" or enlarge its orbit?

Metonic Cycle and Golden Number G

Meton's discovery, announced at the Olympic Games in Athens, generated so much enthusiasm that his formula was engraved in gold in a public place.  The successive numbers from one to nineteen, henceforth known as golden numbers G, identify the individual years of each separate cycle.  His designation of seven values of G (3, 6, 8, 11, 14, 17 and 19) as leap years--when a thirteenth month containing 29 days is added to the 12, 29, and 30-day months--is known as the Metonic cycle.  The present Hebrew calendar year 5735, for instance, corresponds to the golden number found as the remainder, after dividing by 19 and ignoring the whole-number portion of the quotient, readily found as G = 16, hence an ordinary year.  Other lunar and solar calendars also follow the Metonic cycle numbering system.

The general form of the adjusted lunar calendar is therefore essentially completed.  The 235 lunar months (or lunations) comprise 12 years of 1 2 months each and 7 years of 13 months.  The assignment of either 29 or 30 days in three particular months provides enough freedom to assure that certain religious holidays of prescribed date do not fall on particular weekdays (Yom Kippur never on Sunday, nor on Friday, etc.).  One New Moon occurring on a given civil date corresponds to another New Moon 19 years later, as well as earlier, on the same civil date as a rule.  The one and a half hour shift in each 19 years and actual variations of lunar months from the mean value by as much as 15 hours may cause occasional one-day discrepancies one way or the other.

The method of continued fractions employed above to demonstrate the 19-year lunar calendar cycle also has various other applications.  One of these is the demonstration of the Saros eclipse cycle of 18 years and 11 days, also based on precise knowledge and constancy of Earth and Moon motions.  A discussion of this cycle is planned for a later issue of Pensée.  Other applications that may be mentioned include the design of gear trains and analysis of electrical networks required to assure clear imaging on television screens.

Author's gratitude is gladly extended to Mr. William D. Weisberg for drawing attention to lunar calendar construction, and for his helpful suggestions and discussion.


*Feldman, W. M. (1965): Rabbinical Mathematics and Astronomy.  Hermon Press, 235 pp.

Merritt, H. E. (1947): Gear Trains.  Pitman, 178 pp.

Rasof, B. (1970): "Continued Fractions and 'Leap' Years," The Mathematics Teacher, vol. 43, No, 1,  pp. 23-27.

Smart, W. M. (1962): Spherical Astronomy, Fifth Edition.  Cambridge, 430 pp.

Smith, D. E., and Ginsburg, J. (1918): "Rabbi Ben Ezra and the Hindu-Arabic Problem," American       Mathematical Monthly, vol. 25, pp. 99-108.

Steinschneider, M. (1870): "Zur Geschichte der Uebersetzungen aus dem Indischen ins Arabische," Zeitschrift der deutschen Morgenlandischen Gesellschaft, Bd. 24, P. 32S.

Velikovsky, I, (1950): Worlds in Collision,  Doubleday and Dell.  Chapter 8.

Weinberg, L. (1962): Network Analysis and Synthesis.  McGraw-Hill.

PENSEE Journal X

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